Uniqueness of Optimal Production Schedule with Convex Costs and Capacity Constrains

This paper revisits the classical production planning problem studied in Modigliani and Hong (1955). In that seminal paper, Modigliani and Hohn solved the problem of satisfying deterministic demands over a finite-planning horizon at the lowest possible cost. The objective was to minimize the sum of production costs and inventory holding costs. They assumed unlimited capacity and strictly increasing continuous marginal cost functions. One of their key results is that the optimal production schedule of a long planning horizon can be decomposed into solutions to a set of problems with shorter planning horizon (thus more tractable). Furthermore, the optimal schedule of each shorter horizon depends on requirements within the time interval itself, and is independent of requirements of earlier or later periods. The result implies that often only a few parameters need to be accurately estimated and forecast for production decisions.

The key contribution of this paper is the uniqueness of optimal production schedule. The uniqueness theorem can not only dramatically simplify the proof of optimality in Modigliani and Hohn (1955), but also allow extension to a broader set of problems (e.g., general convex production cost functions with capacity constraints in this paper). Moreover, the permutation method used in this paper, to our best knowledge, is the first of its kind in the literature for production planing. Permutation method is a commonly applied mathematical trick. When applied to solve problems of production planning, the method is a useful tool in proving optimality results with a variety of problem settings.